New submissions for Fri, 5 Jun 20

In the paper, we experimentally study the inverse problem with the resonant
scattering determinant. We analyze the structure of characteristics of
perturbed linear waves. Assuming there is the common part of potential
perturbation propagating along the same strips, we estimate the common part of
the perturbed wave, and its Fourier transform.
We deduce the partial inverse uniqueness from the Nevanlinna type of
representation theorem.

Let $L^2(D)$ be the space of measurable square-summable functions on the unit
disk. Let $L^2_a(D)$ be the Bergman space, i.e., the (closed) subspace of
analytic functions in $L^2(D)$. $P_+$ stays for the orthogonal projection going
from $L^2(D)$ to $L^2_a(D)$. For a function $\varphi\in L^\infty(D)$, the
Toeplitz operator $T_\varphi: L^2_a(D)\to L^2_a(D)$ is defined as $$ T_\varphi
f=P_+\varphi f, \quad f\in L^2_a(D). $$ The main result of this article are
spectral asymptotics for singular (or eigen-) values of compact Toeplitz
operators with logarithmically decaying symbols, that is $$
\varphi(z)=\varphi_1(e^{i\theta})\, (1+\log(1/(1-r)))^{-\gamma},\quad \gamma>0,
$$ where $z=re^{i\theta}$ and $\varphi_1$ is a continuous (or piece-wise
continuous) function on the unit circle. The result is applied to the spectral
analysis of banded (including Jacobi) matrices.

In this work we prove that the eigenvalues of the $n$-dimensional massive
Dirac operator $\mathscr{D}_0 + V$, $n\ge2$, perturbed by a possibly
non-Hermitian potential $V$, are localized in the union of two disjoint disks
of the complex plane, provided that $V$ is sufficiently small with respect to
the mixed norms $L^1_{x_j} L^\infty_{\widehat{x}_j}$, for $j\in\{1,\dots,n\}$.
In the massless case, we prove instead that the discrete spectrum is empty
under the same smallness assumption on $V$, and in particular the spectrum is
the same of the unperturbed operator, namely
$\sigma(\mathscr{D}_0+V)=\sigma(\mathscr{D}_0)=\mathbb{R}$. The main tools we
employ are an abstract version of the Birman-Schwinger principle, which include
also the study of embedded eigenvalues, and suitable resolvent estimates for
the Schr\"odinger operator.

In this paper we first prove a general result about the uniqueness and
non-degeneracy of positive radial solutions to equations of the form $\Delta
u+g(u)=0$. Our result applies in particular to the double power non-linearity
where $g(u)=u^q-u^p-\mu u$ for $p>q>1$ and $\mu>0$, which we discuss with more
details. In this case, the non-degeneracy of the unique solution $u_\mu$ allows
us to derive its behavior in the two limits $\mu\to0$ and $\mu\to\mu_*$ where
$\mu_*$ is the threshold of existence. This gives the uniqueness of energy
minimizers at fixed mass in certain regimes. We also make a conjecture about
the variations of the $L^2$ mass of $u_\mu$ in terms of $\mu$, which we
illustrate with numerical simulations. If valid, this conjecture would imply
the uniqueness of energy minimizers in all cases and also give some important
information about the orbital stability of $u_\mu$.

Our goal is to find an asymptotic behavior as $n\to\infty$ of the orthogonal
polynomials $P_{n}(z)$ defined by Jacobi recurrence coefficients $a_{n}$
(off-diagonal terms) and $ b_{n}$ (diagonal terms). We consider the case
$a_{n}\to\infty$, $b_{n}\to\infty$ in such a way that $\sum a_{n}^{-1}<\infty$
$($that is, the Carleman condition is violated$)$ and $\gamma_{n}:=2^{-1}b_{n}
(a_{n}a_{n-1})^{-1/2} \to \gamma $ as $n\to\infty$. In the case $|\gamma | \neq
1$ asymptotic formulas for $P_{n}(z)$ are known; they depend crucially on the
sign of $| \gamma |-1$. We study the critical case $| \gamma |=1$. The formulas
obtained are qualitatively different in the cases $|\gamma_{n}|
\to 1-0$ and $|\gamma_{n}|
\to 1+0$. Another goal of the paper is to advocate an approach to a study of
asymptotic behavior of $P_{n}(z)$ based on a close analogy of the Jacobi
difference equations and differential equations of Schr\"odinger type.